GRIDLESS METHOD FOR UNSTEADY VISCOUS FLOWS

Gridless method is developed for unsteady viscous flows involving moving boundaries. The point distri- bution of gridless method is implemented in an isotropic or anisotropic way according to the features of viscous flows. In the area far away from the body, the traditional cloud of isotropic points is used, while in the adjacent area, the cloud of anisotropic points is distributed. In this way, the point spacing normal to the wall can be small enough for simulating the boundary layer, and meanwhile, the total number of points in the computational do- main can be controlled due to large spacing in other tangential direction through the anisotropic way. A fast mov- ing technique of clouds of points at each time-step is presented based on the attenuation law of disturbed motion for unsteady flows involving moving boundaries. In the mentioned cloud of points, a uniform weighted least- square curve fit method is utilized to discretize the spatial derivatives of the Navier-Stokes equations. The pro- posed gridless method, coupled with a dual time-stepping method and the Spalart-Allmaras turbulence model, is implemented for the Navier-Stokes equations. The computational results of unsteady viscous flows around a NLR7301 airfoil with an oscillating flap and a pitching NACA0012 airfoil are presented in a good agreement with the available experimental data.

A Preconditioned Gridless Method for Solving Euler Equations at Low Mach Numbers

A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weiss and Smith to the time derivative of the Euler equations,which are discretized using agridless technique wherein the physical domain is distributed by clouds of points.The implementation of the preconditioned gridless method is mainly based on the frame of the traditional gridless method without preconditioning,which may fail to converge for low Mach number simulations.Therefore,the modifications corresponding to the affected terms of preconditioning are mainly addressed.The numerical results show that the preconditioned gridless method still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The paper ends with the nearly incompressible flow over a multi-element airfoil,which demonstrates the ability of the method presented for treating flows over complicated geometries.

Preconditioned gridless methods for solving three-dimensional Euler equations at low Mach numbers

In this paper,preconditioned gridless methods are developed for solving the threedimensional(3D)Euler equations at low Mach numbers.The preconditioned system is obtained by multiplying a preconditioning matrix of the type of Weiss and Smith to the time derivative of the 3D Euler equations,which are discretized under the clouds of points distributed in the computational domain by using a gridless technique.The implementations of the preconditioned gridless methods are mainly based on the frame of the traditional gridless method without preconditioning,which may fail to have convergence for flow simulations at low Mach numbers,therefore the modifications corresponding to the affected terms of preconditioning are mainly addressed in the paper.An explicit four-stage Runge–Kutta scheme is first applied for time integration,and the lower-upper symmetric Gauss-Seidel(LU-SGS)algorithm is then introduced to form the implicit counterpart to have the further speed up of the convergence.Both the resulting explicit and implicit preconditioned gridless methods are validated by simulating flows over two academic bodies like sphere or hemispherical headform,and transonic and nearly incompressible flows over one aerodynamic ONERA M6 wing.The gridless clouds of both regular and irregular points are used in the simulations,which demonstrates the ability of the method presented for coping with flows over complicated aerodynamic geometries.Numerical results of surface pressure distributions agree well with available experimental data or simulated solutions in the literature.The numerical results also show that the preconditioned gridless methods presented still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The convergence of the implicit preconditioned gridless method,as expected,is much faster than its explicit counterpart.